Problem: $f(x) = \sqrt{ 8 - \lvert x \rvert }$ What is the domain of the real-valued function $f(x)$ ?
$f(x)$ is undefined when the radicand (the expression under the radical) is less than zero. So we know that $8 - \lvert x \rvert \geq 0$ So $\lvert x \rvert \leq 8$ This means $x \leq 8$ and $x \geq -8$ ; or, equivalently, $-8 \leq x \leq 8$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid -8\leq x \leq8\, \}$.